Subgroups — C₂²×C₃₁×#359 #21490142

Order 7936.

1174 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#135	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#359	C₂	1	12	yes	maximal, direct product factor
5	C₂²×#41×C₃₁	—	2	1	no	—
6	C₂×C₃₁×#135	—	2	3	no	—
7	C₂×C₃₁×#135	—	2	3	no	—
8	C₂×C₃₁×#135	C₄	1	6	yes	—
9	C₂×C₃₁×#135	C₂²	1	6	yes	—
10	C₂²×Q₁₆×C₃₁	C₄	1	1	yes	characteristic
11	C₂²×SD₁₆×C₃₁	—	2	1	no	—
12	C₂²×D₈×C₃₁	C₄	1	1	yes	characteristic
13	C₂²×M₁₆×C₃₁	—	2	1	no	—
14	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₃₁	—	2	1	no	—
16	C₂²×C₁₆×C₃₁	—	2	3	no	—
17	C₂²×C₁₆×C₃₁	C₂²	1	6	yes	complemented
18	C₂×C₃₁×#108	—	2	3	no	—
19	C₂×C₃₁×#108	C₂²	1	6	yes	complemented
20	C₃₁×#359	C₂²	1	16	yes	direct product factor
21	C₂×#41×C₃₁	—	2	6	no	—
22	C₂×#41×C₃₁	—	2	6	no	—
23	C₂×#41×C₃₁	—	4	3	no	—
24	C₂×#41×C₃₁	—	4	3	no	—
25	C₂²×Q₈×C₃₁	—	2	1	no	—
26	C₂²×D₄×C₃₁	—	2	1	no	—
27	C₂²×D₄×C₃₁	—	4	1	no	—
28	C₂³×C₄×C₃₁	—	4	1	no	—
29	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
30	C₃₁×#135	—	2	12	no	—
31	C₃₁×#135	—	2	6	no	—
32	C₃₁×#135	—	2	6	no	—
33	C₃₁×#135	C₂×C₄	1	12	yes	—
34	C₃₁×#135	C₂³	1	4	yes	—
35	C₂×Q₁₆×C₃₁	—	2	3	no	—
36	C₂×Q₁₆×C₃₁	C₂×C₄	1	6	yes	—
37	C₂×SD₁₆×C₃₁	—	2	6	no	—
38	C₂×SD₁₆×C₃₁	—	2	6	no	—
39	C₂×D₈×C₃₁	—	2	3	no	—
40	C₂×D₈×C₃₁	C₂×C₄	1	6	yes	—
41	C₂×M₁₆×C₃₁	—	2	6	no	—
42	C₂×M₁₆×C₃₁	—	4	3	no	—
43	C₂²×C₈×C₃₁	—	4	1	no	—
44	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	C₂×C₄	1	3	yes	—
46	C₂²×C₈×C₃₁	D₄	1	3	yes	—
47	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₃₁	D₄	1	3	yes	—
49	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
50	C₂×C₁₆×C₃₁	—	2	6	no	—
51	C₂×C₁₆×C₃₁	—	2	6	no	—
52	C₂×C₁₆×C₃₁	C₂³	1	4	yes	complemented
53	C₃₁×#108	—	2	6	no	—
54	C₃₁×#108	C₂³	1	4	yes	complemented
55	C₂⁴×C₃₁	—	4	1	no	—
56	C₂⁴×C₃₁	SD₁₆	1	1	yes	characteristic
57	#41×C₃₁	—	2	12	no	—
58	#41×C₃₁	—	2	4	no	—
59	#41×C₃₁	—	4	12	no	—
60	#41×C₃₁	—	4	6	no	—
61	#41×C₃₁	—	4	6	no	—
62	C₂×Q₈×C₃₁	—	2	6	no	—
63	C₂×Q₈×C₃₁	—	4	3	no	—
64	C₂×D₄×C₃₁	—	2	6	no	—
65	C₂×D₄×C₃₁	—	4	6	no	—
66	C₂×D₄×C₃₁	—	4	6	no	—
67	C₂×D₄×C₃₁	—	4	3	no	—
68	C₂²×C₄×C₃₁	—	4	1	no	—
69	C₂²×C₄×C₃₁	—	4	6	no	—
70	C₂²×C₄×C₃₁	—	4	6	no	—
71	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	C₂²⋊C₄	1	3	yes	—
73	C₂²×C₄×C₃₁	D₈	1	3	yes	—
74	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	SD₁₆	1	3	yes	—
76	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
77	Q₁₆×C₃₁	—	2	6	no	—
78	Q₁₆×C₃₁	C₂²×C₄	1	4	yes	—
79	SD₁₆×C₃₁	—	2	12	no	—
80	SD₁₆×C₃₁	—	2	4	no	—
81	D₈×C₃₁	—	2	6	no	—
82	D₈×C₃₁	C₂²×C₄	1	4	yes	—
83	M₁₆×C₃₁	—	2	4	no	—
84	M₁₆×C₃₁	—	4	6	no	—
85	C₂×C₈×C₃₁	—	4	6	no	—
86	C₂×C₈×C₃₁	C₂²⋊C₄	1	3	yes	—
87	C₂×C₈×C₃₁	C₂²⋊C₄	1	3	yes	—
88	C₂×C₈×C₃₁	C₂²⋊C₄	1	6	yes	—
89	C₂×C₈×C₃₁	C₂²×C₄	1	3	yes	—
90	C₂×C₈×C₃₁	C₂²×C₄	1	3	yes	—
91	C₂×C₈×C₃₁	C₂×D₄	1	3	yes	—
92	C₂×C₈×C₃₁	C₂×D₄	1	3	yes	—
93	C₂×C₈×C₃₁	C₂×D₄	1	3	yes	—
94	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
95	C₁₆×C₃₁	—	2	4	no	—
96	C₂²×#359	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₃₁	—	2	4	no	—
98	C₂³×C₃₁	—	4	6	no	—
99	C₂³×C₃₁	—	8	4	no	—
100	C₂³×C₃₁	#102	1	3	yes	—
101	C₂³×C₃₁	#102	1	1	yes	characteristic
102	C₂³×C₃₁	C₂×SD₁₆	1	3	yes	—
103	Q₈×C₃₁	—	2	4	no	—
104	Q₈×C₃₁	—	4	6	no	—
105	D₄×C₃₁	—	2	4	no	—
106	D₄×C₃₁	—	4	4	no	—
107	D₄×C₃₁	—	4	12	no	—
108	D₄×C₃₁	—	4	6	no	—
109	C₂×C₄×C₃₁	—	4	6	no	—
110	C₂×C₄×C₃₁	—	4	4	no	—
111	C₂×C₄×C₃₁	—	4	12	no	—
112	C₂×C₄×C₃₁	#102	1	3	yes	—
113	C₂×C₄×C₃₁	#102	1	3	yes	—
114	C₂×C₄×C₃₁	#102	1	6	yes	—
115	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	3	yes	—
116	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	3	yes	—
117	C₂×C₄×C₃₁	C₂×D₈	1	3	yes	—
118	C₂×C₄×C₃₁	C₂×D₈	1	3	yes	—
119	C₂×C₄×C₃₁	C₂×SD₁₆	1	3	yes	—
120	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
121	C₈×C₃₁	—	4	4	no	—
122	C₈×C₃₁	C₂×C₂²⋊C₄	1	3	yes	—
123	C₈×C₃₁	C₂×C₂²⋊C₄	1	3	yes	—
124	C₈×C₃₁	C₂³×C₄	1	1	yes	characteristic
125	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
126	C₂²×#135	C₂×C₃₁	1	1	yes	characteristic
127	C₂³×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
128	C₂²×#108	C₂×C₃₁	1	1	yes	characteristic, complemented
129	C₂×#359	C₂×C₃₁	1	12	yes	direct product factor
130	C₂²×C₃₁	—	2	6	no	—
131	C₂²×C₃₁	—	2	6	no	—
132	C₂²×C₃₁	—	4	4	no	—
133	C₂²×C₃₁	—	8	12	no	—
134	C₂²×C₃₁	#359	1	4	yes	direct product factor
135	C₂²×C₃₁	C₂×#102	1	3	yes	—
136	C₂²×C₃₁	C₂×#102	1	3	yes	—
137	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
138	C₄×C₃₁	—	4	4	no	—
139	C₄×C₃₁	C₂×#102	1	3	yes	—
140	C₄×C₃₁	C₂×#102	1	3	yes	—
141	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
142	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
143	C₂²×#41	—	2	1	no	—
144	C₂×#135	—	2	3	no	—
145	C₂×#135	—	2	3	no	—
146	C₂×#135	C₄×C₃₁	1	6	yes	—
147	C₂×#135	C₂²×C₃₁	1	6	yes	—
148	C₂²×Q₁₆	C₄×C₃₁	1	1	yes	characteristic
149	C₂²×SD₁₆	—	2	1	no	—
150	C₂²×D₈	C₄×C₃₁	1	1	yes	characteristic
151	C₂²×M₁₆	—	2	1	no	—
152	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
153	C₂²×C₁₆	—	2	1	no	—
154	C₂²×C₁₆	—	2	3	no	—
155	C₂²×C₁₆	C₂²×C₃₁	1	6	yes	complemented
156	C₂×#108	—	2	3	no	—
157	C₂×#108	C₂²×C₃₁	1	6	yes	complemented
158	#359	C₂²×C₃₁	1	16	yes	direct product factor
159	C₂×C₃₁	—	2	1	no	—
160	C₂×C₃₁	—	2	3	no	—
161	C₂×C₃₁	—	8	4	no	—
162	C₂×C₃₁	C₂×#359	1	6	yes	direct product factor
163	C₂×C₃₁	C₂²×#102	1	1	yes	characteristic
164	C₂×#41	—	2	6	no	—
165	C₂×#41	—	2	6	no	—
166	C₂×#41	—	4	3	no	—
167	C₂×#41	—	4	3	no	—
168	C₂²×Q₈	—	2	1	no	—
169	C₂²×D₄	—	2	1	no	—
170	C₂²×D₄	—	4	1	no	—
171	C₂³×C₄	—	4	1	no	—
172	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
173	#135	—	2	12	no	—
174	#135	—	2	6	no	—
175	#135	—	2	6	no	—
176	#135	C₂×C₄×C₃₁	1	12	yes	—
177	#135	C₂³×C₃₁	1	4	yes	—
178	C₂×Q₁₆	—	2	3	no	—
179	C₂×Q₁₆	C₂×C₄×C₃₁	1	6	yes	—
180	C₂×SD₁₆	—	2	6	no	—
181	C₂×SD₁₆	—	2	6	no	—
182	C₂×D₈	—	2	3	no	—
183	C₂×D₈	C₂×C₄×C₃₁	1	6	yes	—
184	C₂×M₁₆	—	2	6	no	—
185	C₂×M₁₆	—	4	3	no	—
186	C₂²×C₈	—	4	1	no	—
187	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
188	C₂²×C₈	C₂×C₄×C₃₁	1	3	yes	—
189	C₂²×C₈	D₄×C₃₁	1	3	yes	—
190	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
191	C₂²×C₈	D₄×C₃₁	1	3	yes	—
192	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
193	C₂×C₁₆	—	2	6	no	—
194	C₂×C₁₆	—	2	6	no	—
195	C₂×C₁₆	C₂³×C₃₁	1	4	yes	complemented
196	#108	—	2	6	no	—
197	#108	C₂³×C₃₁	1	4	yes	complemented
198	C₃₁	C₂²×#359	1	1	yes	characteristic, direct product factor, Sylow
199	C₂⁴	—	4	1	no	—
200	C₂⁴	SD₁₆×C₃₁	1	1	yes	characteristic
201	#41	—	2	12	no	—
202	#41	—	2	4	no	—
203	#41	—	4	12	no	—
204	#41	—	4	6	no	—
205	#41	—	4	6	no	—
206	C₂×Q₈	—	2	6	no	—
207	C₂×Q₈	—	4	3	no	—
208	C₂×D₄	—	2	6	no	—
209	C₂×D₄	—	4	6	no	—
210	C₂×D₄	—	4	6	no	—
211	C₂×D₄	—	4	3	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	6	no	—
214	C₂²×C₄	—	4	6	no	—
215	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊C₄×C₃₁	1	3	yes	—
217	C₂²×C₄	D₈×C₃₁	1	3	yes	—
218	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
219	C₂²×C₄	SD₁₆×C₃₁	1	3	yes	—
220	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
221	Q₁₆	—	2	6	no	—
222	Q₁₆	C₂²×C₄×C₃₁	1	4	yes	—
223	SD₁₆	—	2	12	no	—
224	SD₁₆	—	2	4	no	—
225	D₈	—	2	6	no	—
226	D₈	C₂²×C₄×C₃₁	1	4	yes	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	4	6	no	—
229	C₂×C₈	—	4	6	no	—
230	C₂×C₈	C₂²⋊C₄×C₃₁	1	3	yes	—
231	C₂×C₈	C₂²⋊C₄×C₃₁	1	3	yes	—
232	C₂×C₈	C₂²⋊C₄×C₃₁	1	6	yes	—
233	C₂×C₈	C₂²×C₄×C₃₁	1	3	yes	—
234	C₂×C₈	C₂²×C₄×C₃₁	1	3	yes	—
235	C₂×C₈	C₂×D₄×C₃₁	1	3	yes	—
236	C₂×C₈	C₂×D₄×C₃₁	1	3	yes	—
237	C₂×C₈	C₂×D₄×C₃₁	1	3	yes	—
238	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
239	C₁₆	—	2	4	no	—
240	C₂³	—	2	4	no	—
241	C₂³	—	4	6	no	—
242	C₂³	—	8	4	no	—
243	C₂³	C₃₁×#102	1	3	yes	—
244	C₂³	C₃₁×#102	1	1	yes	characteristic
245	C₂³	C₂×SD₁₆×C₃₁	1	3	yes	—
246	Q₈	—	2	4	no	—
247	Q₈	—	4	6	no	—
248	D₄	—	2	4	no	—
249	D₄	—	4	12	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	6	no	—
252	C₂×C₄	—	4	6	no	—
253	C₂×C₄	—	4	12	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	C₃₁×#102	1	3	yes	—
256	C₂×C₄	C₃₁×#102	1	3	yes	—
257	C₂×C₄	C₃₁×#102	1	6	yes	—
258	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	3	yes	—
259	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	3	yes	—
260	C₂×C₄	C₂×D₈×C₃₁	1	3	yes	—
261	C₂×C₄	C₂×D₈×C₃₁	1	3	yes	—
262	C₂×C₄	C₂×SD₁₆×C₃₁	1	3	yes	—
263	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
264	C₈	—	4	4	no	—
265	C₈	C₂×C₂²⋊C₄×C₃₁	1	3	yes	—
266	C₈	C₂×C₂²⋊C₄×C₃₁	1	3	yes	—
267	C₈	C₂³×C₄×C₃₁	1	1	yes	characteristic
268	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
269	C₂²	—	2	6	no	—
270	C₂²	—	2	6	no	—
271	C₂²	—	4	4	no	—
272	C₂²	—	8	12	no	—
273	C₂²	C₃₁×#359	1	4	yes	direct product factor
274	C₂²	C₂×C₃₁×#102	1	3	yes	—
275	C₂²	C₂×C₃₁×#102	1	3	yes	—
276	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
277	C₄	—	4	4	no	—
278	C₄	C₂×C₃₁×#102	1	3	yes	—
279	C₄	C₂×C₃₁×#102	1	3	yes	—
280	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
281	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
282	C₂	—	2	3	no	—
283	C₂	—	2	1	no	—
284	C₂	—	8	4	no	—
285	C₂	C₂×C₃₁×#359	1	6	yes	direct product factor
286	C₂	C₂²×C₃₁×#102	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	D₈
commutator	C₈	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂³×C₃₁	#102
prime radical	C₂²×C₃₁×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂³×C₁₆×C₃₁	C₂	semidirect
3. C₂²×C₃₁×#108	C₂	semidirect
4. C₂×C₃₁×#359	C₂	direct (N × H)
17. C₂²×C₁₆×C₃₁	C₂²	semidirect
19. C₂×C₃₁×#108	C₂²	semidirect
20. C₃₁×#359	C₂²	direct (N × H)
52. C₂×C₁₆×C₃₁	C₂³	semidirect
54. C₃₁×#108	C₂³	semidirect
96. C₂²×#359	C₃₁	direct (N × H)
127. C₂³×C₁₆	C₂×C₃₁	semidirect
128. C₂²×#108	C₂×C₃₁	semidirect
129. C₂×#359	C₂×C₃₁	direct (N × H)
134. C₂²×C₃₁	#359	direct (N × H)
155. C₂²×C₁₆	C₂²×C₃₁	semidirect
157. C₂×#108	C₂²×C₃₁	semidirect
158. #359	C₂²×C₃₁	direct (N × H)
162. C₂×C₃₁	C₂×#359	direct (N × H)
195. C₂×C₁₆	C₂³×C₃₁	semidirect
197. #108	C₂³×C₃₁	semidirect
198. C₃₁	C₂²×#359	direct (N × H)
273. C₂²	C₃₁×#359	direct (N × H)
285. C₂	C₂×C₃₁×#359	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃₁×#135	2
C₂³×C₁₆×C₃₁	2
C₂²×C₃₁×#108	2
C₂×C₃₁×#359	2
C₂²×#359	31

Subgroups — C22×C31×#359 #21490142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₁×#359 #21490142